J. Aust. Math. Soc.  81 (2006), 153-164
Cubic symmetric graphs of order twice an odd prime-power

Yan-Quan Feng
  Department of Mathematics
  Beijing Jiaotong University
  Beijing 100044
  P.R. China
 
and
Jin Ho Kwak
  Combinatorial and Computational
  Mathematics Center
  Pohang University of Science and Technology
  Pohang, 790--784
  Korea
  jinkwak@postech.ac.kr


Abstract
An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p we show that if $p\not=5,7$ then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if $p\not=3$ then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s – 1)-regular subgroup for each $1\leq s\leq 5$. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each $1\leq s\leq 5$ and each prime p, as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.
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